Determine how many solutions exist for the system of equations. ${2x-2y = 10}$ ${-x+y = -5}$
Answer: Convert both equations to slope-intercept form: ${2x-2y = 10}$ $2x{-2x} - 2y = 10{-2x}$ $-2y = 10-2x$ $y = -5+x$ ${y = x-5}$ ${-x+y = -5}$ $-x{+x} + y = -5{+x}$ $y = -5+x$ ${y = x-5}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = x-5}$ ${y = x-5}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${2x-2y = 10}$ is also a solution of ${-x+y = -5}$, there are infinitely many solutions.